Extreme quantiles of unobserved individual heterogeneity (UIH) are often of interest in analysis of economic panel data and in random effects meta-analysis; UIH includes heterogeneous coefficients, treatment effects, etc. For example, in the setting of [1]} we might be interested in estimating the lowest level of firm productivity that is compatible with firm survival in a given area. This corresponds to finding
the zeroth quantile of the distribution of productivity of surviving firms.
In another example, in random effects meta-analysis, the true treatment effects of the same intervention are allowed to differ between studies. A 95% prediction interval is an interval such that the true treatment effect in a new study falls with probability 95% in it. Such intervals provide an important summary of between-study heterogeneity [2]}, [3]}. To construct such an interval, we need to estimate and quantify uncertainty about the 2.5th and 97.5th percentiles of the distribution of study-specific treatment effects; these percentiles may be relatively extreme for datasets of results of individual studies when number of studies is not large.
